Puzzle of the Day

A lattice is a mathematical structure composed of a set
S and a relation r. If x, y, and z
are elements of S, and r is a relation over those elements,
then (S, r) is a lattice if and only if the following
conditions are met:

r is reflexive, so for every element x in S,
xrx.

r is antisymmetric, so if xry and yrx,
then x = y.

r is transitive, so if xry and yrz,
then xrz.

Every pair of elements in S has a lower bound.
In other words, there is a z in S
such that zrx and zry for every pair (x, y).

Every pair of elements in S has an upper bound.
In other words, there is a z in S
such that xrz and yrz for every pair (x, y).